Treatment vs. effect contrasts

Abstract

Here is a simple example to show the differences between treatment and effect contrasts.

Example dataset

Create a random dataset with criteria y, predictors x1, x2 and gender.

• y and gender are correlated
• y and x1 are correlated only if gender is 1
• y and x2 are correlated only if gender is 1
• x1 and x2 are correlated
• x1 and x2 have an interaction effect on y only if gender is 1

set.seed(1234)
n <- 2000
gender <- rep(0:1, each = n/2)
y <- sample(0:10, n, replace = TRUE) + gender * sample(0:10, n, replace = TRUE)
x1 <- sample(0:10, n, replace = TRUE) + gender * y
x2 <- x1 + sample(0:10, n, replace = TRUE) + gender * y
y <- y + (x1 > median(x1) & x2 > median(x2) & gender == 1) * sample(0:10, n, replace = TRUE) * 2

dat <- data.frame(y = y, x1 = x1, x2 = x2, gender = gender)

Descriptives

psych::corr.test(dat)

Call:psych::corr.test(x = dat)
Correlation matrix 
          y   x1   x2 gender
y      1.00 0.79 0.82   0.68
x1     0.79 1.00 0.95   0.74
x2     0.82 0.95 1.00   0.79
gender 0.68 0.74 0.79   1.00
Sample Size 
[1] 2000
Probability values (Entries above the diagonal are adjusted for multiple tests.) 
       y x1 x2 gender
y      0  0  0      0
x1     0  0  0      0
x2     0  0  0      0
gender 0  0  0      0

 To see confidence intervals of the correlations, print with the short=FALSE option

psych::describe(dat)

	vars	n	mean	sd	median	trimmed	mad	min	max	range	skew	kurtosis	se
y	1	2000	11.6735	9.829965	8.0	10.58375	7.4130	0	40	40	0.8543788	-0.4209326	0.2198047
x1	2	2000	10.0920	6.802549	9.0	9.67625	7.4130	0	30	30	0.5169160	-0.5958746	0.1521096
x2	3	2000	20.2970	12.883508	17.0	19.30562	13.3434	0	58	58	0.6054280	-0.6904413	0.2880840
gender	4	2000	0.5000	0.500125	0.5	0.50000	0.7413	0	1	1	0.0000000	-2.0009998	0.0111831

Contrasts

The left part of the table is with gender as treatment contrast (0 vs. 1) and the right part with gender as effect contrast (-1 vs. 1)

# Gender has values 0 vs. 1 (treatment contrast)
fit1 <- lm(y ~ gender * x1 * x2, data = dat)

# Gender hast -1 vs. 1 (effect contrast)
dat$gender <- car::recode(dat$gender, "0 = -1; 1 = 1")

fit2 <- lm(y ~ gender * x1 * x2, data = dat)

sjPlot::tab_model(fit1, fit2, show.se = TRUE, show.ci = FALSE, col.order = c("est", "se", "std.est", "p"), digits = 4, dv.labels = c("Treatment contrast<br> for gender", "Effect contrast<br> for gender"))

	Treatment contrast for gender			Effect contrast for gender
Predictors	Estimates	std. Error	p	Estimates	std. Error	p
(Intercept)	5.2577	0.6189	<0.001	-4.6274	0.6427	<0.001
gender	-19.7703	1.2853	<0.001	-9.8852	0.6427	<0.001
x1	-0.0695	0.1358	0.609	0.6640	0.0851	<0.001
x2	-0.0270	0.0795	0.734	0.4363	0.0482	<0.001
gender × x1	1.4671	0.1702	<0.001	0.7335	0.0851	<0.001
gender × x2	0.9267	0.0963	<0.001	0.4633	0.0482	<0.001
x1 × x2	0.0056	0.0116	0.629	-0.0124	0.0059	0.036
(gender × x1) × x2	-0.0360	0.0118	0.002	-0.0180	0.0059	0.002
Observations	2000			2000
R2 / R2 adjusted	0.738 / 0.737			0.738 / 0.737

• The intercept in model1 (treatment contrast) is the mean of y for gender 0
• The intercept in model2 (effect contrast) is the mean of y for all data
• All predictors in model1 without a gender term are effects for gender 0 while the interactions with a gender term are effects for gender 1
• All predictors in model2 without a gender term are effects across gender while the interactions with a gender term are the effects of gender (subtracted for gender 0 and added for gender 1).
• gender has a significant effect on y in both models
• x1, x2 and x1*x2 only have a significant effect on y in model 2
• All gender interactions are significant in both models where the effect sizes and the standard errors of model2 are half of the corresponding values in model1, so p is identical.